Math Core

Sometimes you need to multiply three numbers, like $3 \times 5 \times 2$ . Which two do you multiply first? Here is a secret: it doesn't matter! You can pick the pair that makes the problem easiest.

Three numbers, two ways

Say you have 3 boxes. Each box holds 5 packs, and each pack has 2 markers. How many markers are there?

Way 1: packs first. There are $3 \times 5 = 15$ packs. Each has 2 markers: $15 \times 2 = 30$ .

Way 2: one box first. One box holds $5 \times 2 = 10$ markers. There are 3 boxes: $3 \times 10 = 30$ .

Both ways give 30 markers. We use parentheses to show which pair to multiply first:

(3 \times 5) \times 2 = 15 \times 2 = 30

3 \times (5 \times 2) = 3 \times 10 = 30

The associative property of multiplication

When you multiply three numbers, you can group them in any way. The answer stays the same.

(3 \times 5) \times 2 = 3 \times (5 \times 2)

Make the problem easy

Way 2 was easier, because $5 \times 2$ makes 10, and multiplying by 10 is simple. Look for a pair that makes a nice number.

$2 \times 5$ makes 10.
$2 \times 2$ makes 4, and $2 \times 3$ makes 6. Small facts are easy too.

You can also combine this with the commutative property. That lets you move the numbers around first. For $5 \times 7 \times 2$ , flip it to $7 \times 5 \times 2$ . Then group: $7 \times (5 \times 2) = 7 \times 10 = 70$ .

Break a factor into two factors

The associative property also helps with a hard fact. You can write a factor as two smaller factors.

For $4 \times 6$ , write 6 as $2 \times 3$ :

4 \times 6 = 4 \times (2 \times 3) = (4 \times 2) \times 3 = 8 \times 3 = 24

Worked example: Group to make ten

Find $2 \times 9 \times 5$ .

Flip the order so 2 and 5 are next to each other: $9 \times 2 \times 5$ . Then group them.

9 \times (2 \times 5) = 9 \times 10 = 90

$2 \times 9 \times 5 = 90$ .

Worked example: Group the small numbers

Find $3 \times 2 \times 4$ .

Group the first two: $(3 \times 2) \times 4 = 6 \times 4 = 24$ .

Or group the last two: $3 \times (2 \times 4) = 3 \times 8 = 24$ .

Either way, the answer is 24.

Worked example: Split a factor

Find $3 \times 8$ by writing 8 as $4 \times 2$ .

3 \times 8 = 3 \times (4 \times 2) = (3 \times 4) \times 2 = 12 \times 2 = 24

$3 \times 8 = 24$ .

Common mistake

When you split a factor, use multiplication, not addition. 8 is $4 \times 2$ , so $3 \times 8 = (3 \times 4) \times 2$ . If you split 8 into $4 + 4$ , that is the distributive property, and you must add the two parts instead.

Tip

Try the grouping two ways. If both give the same answer, you can be sure you're right.

Practice

Practice 1

Find $4 \times 5 \times 2$ .

Type a number.

Practice 2

Find $2 \times 3 \times 2$ .

Type a number.

Practice 3

Which is equal to $(6 \times 2) \times 5$ ?

Practice 4

Find $5 \times 8 \times 2$ .

Type a number.

Practice 5

Find $3 \times 3 \times 4$ .

Type a number.

Practice 6

Fill in the missing number: $(7 \times 2) \times 4 = \underline{\quad} \times (2 \times 4)$ .

Type a number.

Practice 7

There are 2 classrooms. Each classroom has 6 tables. Each table has 5 chairs. How many chairs are there in all?

Type a number.